coverage in wsns (sweep coverage on graph)...sweep coverage sweep coverage let u = {u 1,u 2,…,u n}...

Coverage in WSNs

(Sweep Coverage on Graph)

Barun Gorain and ParthaPartha SarathiSarathi MandalMandal

Department of Mathematics

Indian Institute of Technology Guwahati

Outline

• Introduction

• Coverage

• Sweep coverage problems

• Previous Results and its Flaw

• Our Solution• Our Solution

• Sweep coverage with different sweep periods and processing time

• Sweep coverage with mobile sensors having different speeds

• Conclusion

2P. S.Mandal, IITG

Introduction

• Coverage it is defined as quality of surveillance of

a sensing function in WSNs.

• Is is a widely studied research area, many efforts

have been made for addressing coverage have been made for addressing coverage

problems in sensor networks, which are:

– Point coverage

– Area coverage,

– Barrier coverage,

– k-coverage, etc.

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Point Coverage

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Area Coverage

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Barrier Coverage

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Sweep Coverage

• For those coverage scenarios, the monitored objective being covered all time, featured as static coverage or full coverage.

• In some applications, patrol inspection/ periodic monitoring are sufficient instead of

• In some applications, patrol inspection/ periodic monitoring are sufficient instead of continuous monitoring, which is featured as a sweep coverage.

• For such applications a small number of mobile sensor nodes can guarantee sweep coveragewith a given sweep period.

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Static coverage/Full coverage

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Sweep Coverage

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Sweep Coverage

Sweep Coverage

Let U = {u1,u2,…,un} be a set of points on a two dimensional plane and M = {m1,m2,…mp} be a set of mobile sensor nodes. A point uiis said to be t-sweep covered if and only if at least one mobile sensor node visits ui within every t time period.

� The set U is said to be globally sweep covered by the mobile sensor nodes of M if all ui are t-sweep covered. sensor nodes of M if all ui are t-sweep covered.

� The time period t is called the sweep period of the points in U.

[1] Sweep coverage with mobile sensors, Mo Li, Wei-Fang Cheng, Kebin Liu, Yunhao Liu, Xiang-Yang Li, and XiangkeLiao, IEEE Trans. Mob. Comput., 10(11):1534–1545, 2011.

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Previous Results

• The contribution of Li et al. in [1] are the

following:

– The sweep coverage problem is NP-hard.

– It is not possible to approximate sweep coverage – It is not possible to approximate sweep coverage

problem with a factor less than 2 unless P =NP.

– Proposed a 3-approximation algorithm for sweep

coverage problem.


Previous Results

Basic idea of the proposed algorithm [1]

• Find an approximated TSP tour among the set of points. (1.5 factor)

• Divide the tour into parts of length (vt/2).• Divide the tour into parts of length (vt/2).

• One mobile sensor is deployed at each of the parts.

• Mobile sensors moves back-and-forth to cover the points belonging to the corresponding parts.


Previous Results


Correctness

• Li et al. proved that the proposed algorithm is a 3 factor approximation algorithm.

• But the statement is wrong

Counter Example:

100

• Number of mobile sensors needed according to [1] is 20 Optimalsolution is 2 as two sensor at two points will be sufficient


100

v=20, t=1

Sweep Coverage Problem

• We introduce a variation of sweep coverage named as GSweep coverage problem, where the PoIs are represented by vertices of a weighted graph.

• We propose a 3−approximation algorithm to guarantee • We propose a 3−approximation algorithm to guarantee sweep coverage of all vertices of the graph.

• We generalize the above algorithm to solve the problem with approximation factor O(log ρ) when vertices of the graph have different sweep periods and processing times, where ρ is the ratio of the max and min sweep periods


GSweep coverage problem

• Let G = (U,E) be a weighted graph, where weight of an edge (ui, uj) for (ui, uj) ∈U is denoted by |(ui, uj)|. Let n be the total number of vertices in G. For any subgraph H of G, we denote |H| as the sum of the edge weights of H.

• Definition (GSweep coverage): Let U = {u1, u2, · · ·, un} be the vertices of a weighted graph G = (U, E, w) and M = {m1,m2, · · · ,mn} vertices of a weighted graph G = (U, E, w) and M = {m1,m2, · · · ,mn} be the set of mobile sensors. The mobile sensors move with a uniform speed v along the edges of the graph. For given t > 0, find the minimum number of mobile sensors such that each vertex of G is t-sweep covered.

• The problem is NP hard, follows from the hardness proof given in [1].

16P. S.Mandal, IITG

Algorithm


13

8

12

6

5

5

4

Example

18P. S.Mandal, IITG

For k=1, The weight of

the Forest F is 22

1st Iteration, vt=8

13

8

12

6

5

5

4

85

5

4

the Forest F1 is 22

Number of sensor

nodes needed is

⌈44/8⌉=6

19P. S.Mandal, IITG


the Forest F is 14

2nd Iteration, vt=8

13

8

12

6

5

5

4

5

5

4

the Forest F2 is 14

Number of sensor

nodes needed is

⌈28/8⌉+1= 5

20P. S.Mandal, IITG


the Forest F is 9

3rd Iteration, vt=8

13

8

12

6

5

5

4

5

5

4

the Forest F3 is 9

Number of sensor

nodes needed is

⌈18/8⌉+2=5

21P. S.Mandal, IITG


the Forest F is 9

4th Iteration, vt=8

13

8

12

6

5

5

4

4

the Forest F4 is 9

Number of sensor

nodes needed is

⌈8/8⌉+3=4

22P. S.Mandal, IITG


5th Iteration, vt=8

13

8

12

6

5

5

4


the Forest F5 is 9

Number of sensor

nodes needed is 5

23P. S.Mandal, IITG

Analysis

Lemma: Let opt be the minimum number of sensors needed in the optimal solution. Let opt' be the minimum number of paths of length ≤ vt which span U on G. Then opt ≥ opt'.

Proof: Let us assume that opt < opt′.

Let P1, P2,…, Popt be the movement paths of the sensors in optimal solution in a time interval [t ,t+t ]. Then |P |≤vt.

Let P1, P2,…, Popt be the movement paths of the sensors in optimal solution in a time interval [t0,t+t0]. Then |Pi|≤vt.

• According to the definition of Gsweep coverage, all vertices of G are covered by the set of paths P1, P2,…, Popt.

• Hence P1, P2,…, Popt is a collection of paths with |Pi| ≤ vt which spans U on G, which contradicts the fact that opt < opt′.

• Therefore opt ≥ opt'.


Analisys

Theorem: The Algorithm is a 3 factor approximation algorithm.

Proof:

• Let opt be the number of mobile sensors needed in optimal solution.in optimal solution.

• Let opt' be the minimum number of paths of length ≤ vt which span U on G.

• We know opt ≥ opt’ (from previous lemma )

• Let Min_path be the sum of the edge weights of opt' number of paths of length ≤ vt which span U.


Proof (continue..)

• Algorithm 1 chooses the minimum over all Nk for k = 1 to n

• N ≤ Nk for k=1 to n.

• Consider the iteration when k=opt’.

• Then |Fk|≤ Min_path and Min_path ≤ k.vt• Then |Fk|≤ Min_path and Min_path ≤ k.vt

• Total length of the tours after doubling the edges of Fkis 2|Fk|.

• Since ⌈|Ti|/vt⌉ ≤ (|Ti|/vt)+1, therefore total number of sensor nodes needed is

Nk= 2(|Fk|/vt) + k ≤ 2k+k = 3k ≤ 3 opt.

Therefore approximation factor of our algorithm is 3.


The Algorithm is a 3 factor

approximation algorithm.


Sweep Coverage with Different Sweep

Periods and Processing Time

• In practice some finite amount of time i.e.,

processing time is required by a mobile sensor

to process some tasks such as monitoring,

sampling or exchanging data at each of the sampling or exchanging data at each of the

vertices during its visits.

• Sweep periods may be different for different

vertices as per requirement of applications.


Sweep Coverage with Different Sweep

Periods and Processing Time

• Problem 2. Let U = {u1, u2, · · ·, un} be the vertex set of a weighted graph G = (U, E, w), {t1, t2, · · · , tn} and {Ʈ1, Ʈ2, · · · , Ʈn} be the corresponding set of sweep periods and processing times of the vertices. Let M = processing times of the vertices. Let M = {m1,m2, · · · ,mn} be the set of mobile sensors which can move with a uniform speed v along the edges of the graph. Find the minimum number of mobile sensors such that each ui is ti-sweep covered.


Proposed Algorithm

• The proposed algorithm execute in two

different phases.

• In the 1st phase, we will compute the number

of mobile sensors required for sweep of mobile sensors required for sweep

coverage.

• In the 2nd phase, initial positions and

movement schedules of mobile sensors are

computed.


Finding Number of Mobile sensors

• For the given graph G=(U,E,w), compute the

complete graph G’=(U, E’,w’), where

w’(ui,uj)=d(ui,uj)+v(Ʈi+Ʈj)/2

Where d(u ,u ) is the shortest path between u ,uWhere d(ui,uj) is the shortest path between ui,uj

with respect to w.

• Let ƪ=(tmax/tmin ), where tmin and tmax are the

minimum and maximum sweep periods

among the vertices.


Finding Number of Mobile sensors

• Let Ui ={uj| 2i-1 tmin≤ tj< 2i tmin }

• {Ui | i= 1 to ⌈log ƪ⌉} is a partition of U.

• Let G’i be the induced subgraph of G’ for the vertex set Ui.

⌈ ⌉• For each i, i= 1 to ⌈log ƪ⌉, Apply step 1 to 12 of Algorithm 1 on G’i to find the number of mobile sensors required for (2i−1 · tmin)-sweep coverage for all the vertex in Ui.


Movement strategy of the mobile

sensors

• Let C1, C2, …., Cjibe the connected

components for G’i.

• Find tours T1, T2, …., Tjiafter doubling the

edges.edges.

• If Tk contains only one vertex, deploy one

mobile sensor which acts like a static sensor.



sensors

• If Tk contains more than one vertex, compute

T’k by replacing vertices and edges of Tk.

• Let ui1,ui2

,…, uihbe the vertices of Tk in the

clockwise direction. clockwise direction.

• Replace each uilby two vertices u’il

and u’’il

and introduce an edge (u’il, u’’il

) with

w’ (u’il, u’’il

) = vƮl.

• Each edge (uil, uil+1

) is replaced by (u’’il, u’il+1

)



sensors



sensors

• Partition T’k into ⌈w’(T’k)/vt⌉ parts of weight at

most vt.

• Deploy one mobile sensor at each of the

partitioning points.partitioning points.

• The mobile sensor starts their movement

along T’k at the same time in the same

direction.


Partition on the tour T’k

• The mobile sensor deployed

at P will wait for a time taken

to move from P to ui1’’. Then

it moves with velocity v along

the tour.

•The Mobile sensor deployed

at Q will start moving along

the edges (ui2,ui3) in G’.


Analysis

• Theorem: According to the movement strategy of the mobile sensors each vertex ui in U is ti-GSweep covered with processing time Ʈi.

Proof:

• If ui belongs to a component with one vertex, then tisweep coverage is trivial by the mobile sensor deployed sweep coverage is trivial by the mobile sensor deployed at ui.

• Now if ui belongs to a component with more than one vertex, then according to the proposed algorithm for Problem 2, ui ∊ Uj for some j = 1 to ⌈ log Ɗ⌉ .

• By Theorem 1, ui is sweep covered with sweep period 2j−1

· tmin ≤ ti. Therefore ui is ti sweep covered.


Analysis

• Theorem: The approximation factor of the

proposed algorithm for Problem 2 is 6 ⌈ log ƪ⌉ .

Proof:

• Let OPT be the optimal solution for problem 2.

⌈ ⌉

• Let OPT be the optimal solution for problem 2.

• Let OPT1, OPT2, …, OPT⌈ log ƪ⌉ be the optimal

solutions on G’1, G’2, …, G’⌈ log ƪ⌉.

• Then OPT ≤ OPTj, for j = 1 to ⌈ log Ɗ⌉


Proof(cont..)

• Let OUT1, OUT2, …, OUT⌈ log ƪ⌉ be the number of mobile sensors required for our algorithm.

• Then, OUTj ≤ 6 OPTj , j = 1 to ⌈ log Ɗ⌉, as as the length of the partition in OPTj is at most twice of the length of the partition in OUTj and the Algorithm 1 is a 3-approximation algorithm.the length of the partition in OUTj and the Algorithm 1 is a 3-approximation algorithm.

• Therefore, total number of mobile sensors required is

• OUT1+OUT2+ …+OUT⌈ log ƪ⌉ ≤ 6 ⌈ log Ɗ⌉ OPT.

• Hence the proof follows.


Sweep Coverage with Mobile Sensors

Having Different Speeds

• Problem 3: Let U = {u1, u2, · · ·, un} be the

vertex set of a weighted graph G = (U, E, w).

Let M = {m1,m2, · · · ,mn} be the set of mobile

sensors with velocities {v1,v2,…, vn} sensors with velocities {v1,v2,…, vn}

respectively. For a given t>0, Find the

minimum number of mobile sensors such that

each ui is t-sweep covered.


Sweep Coverage with Mobile Sensors

Having Different Speeds

• Theorem: No polynomial time constant factor

approximation algorithm exists to solve the

sweep coverage problem by mobile sensors

with different velocities unless P=NP.with different velocities unless P=NP.

Proof :

We will prove this Theorem using reduction

from Metric-TSP problem.


Proof(Cont..)

• If possible let there is a k factor approximation

algorithm A.

• Consider an instance (G1,L) of metric TSP

problem, where L is weight of the minimum problem, where L is weight of the minimum

weight TSP tour in a complete weighted graph

G1 = (U1, E1, w1) with n (> k) Vertices which

satisfy triangular inequality.


Proof(Cont..)

• Construct a complete graph G2=(U2,E2,w2)

with n2 vertices as follows:

For each vertex ui in U1, consider n vertices For each vertex ui in U1, consider n vertices

ui1,ui2,…, uin in U2.

w2(uik,ujl)=w1(ui,uj)-1/(n+1)2

w2(uik,uil)=1/(n-1)(n+1)2


Proof(Cont..)

• Claim: G1 has a tour with weight at most l iff

G2 has a tour with weight at most l.

Let T1: ui1,ui2,…, uin ,ui1 be a tour of G with

weight ≤ l.weight ≤ l.

Construct T2: ui11,ui1

2,…ui1n,ui2

1, ui22,ui2

n,…uin1,

uin2,…uin

n ,ui11.

Note that w2(T2) ≤ l.


Proof(Cont..)

• Claim: G1 has a tour with weight at most l iff G2 has a tour with weight at most l.

Conversely, let T2’: x1,x2,…,xn2, x1 be a tour in G2 with w2(T2’) ≤ l.

• Construct a tour T1’ from T2’ as follows:• Construct a tour T1’ from T2’ as follows:

Delete all the edges of type (uik,uil) from T2’.

For each of the remaining edges of form (uij,uki), consider the corresponding edge in G1 and construct the sub graph of G1.

• For example, if edges (uk i , ul j ) and (uk2 i , ul2 j ) for k1 , k2 and l1 , l2 are in T0 2, then we consider the edge (ui, uj) twice in E


Proof(Cont..)

• Claim: G1 has a tour with weight at most l iff G2 has a tour with weight at most l.

• Construct a tour T1’ from E by short cutting.

• Note that, w1(T1’) ≤l• Note that, w1(T1’) ≤l


Proof(Cont..)

• Hence we can say that if L is the weight of the

optimal tour in G1, then the weight of the

optimal tour in G2 is also L, otherwise by the

above fact, a tour of G1 with weight less than above fact, a tour of G1 with weight less than

L can be found.


Proof(Cont..)

• We consider an instance of the sweep coverage problem by mobile sensors with different velocities as follows:

• We take G2 as the graph, sweep period t = 1, and a set of n2 mobile sensors with velocity L for one a set of n2 mobile sensors with velocity L for one and zero for remaining n2 − 1 mobile sensors.

• Clearly, the optimal solution of the problem is one, since using the mobile sensor with velocity L, 1-sweep coverage of each of n2 vertices can be guaranteed.


Proof(Cont..)

• As A is a k factor approximation algorithm, A returns

• the number of mobile sensors required for sweep coverage of G2 is k in worst case.

• Among these k mobile sensors, one is with • Among these k mobile sensors, one is with velocity L and remaining are with velocities zero.

• Therefore, the mobile sensor with velocity L moves along a tour covering n2 − k + 1 vertices.

• Remaining k − 1 mobile sensors cover remaining

k − 1 vertices one for each.


Proof(Cont..)

• Hence at least one vertex from each of the set

{u1i , u2i, · · · , uni} for different i is visited by

the mobile sensor with velocity L and weight

of the tour is at most L.of the tour is at most L.

• From this tour, a tour T of weight at most L of

G1 can be constructed in the same way by

constructing an Eulerian graph of G1 and short

cutting as explained before.


Proof(Cont..)

• Since L is the weight of the optimal tour in G1,

w(T) = L.

• Hence optimal solution for TSP problem on G1

can be computed in polynomial time by can be computed in polynomial time by

applying algorithm A on G2, which is not

possible unless P=NP.

• Hence statement of the theorem follows.


Conclusion

• In this paper we overcome the limitation of a previous study [1] on sweep coverage.

• The key argument is that when the graph is sparse, it is better to provide sweep coverage with a mixture of static and mobile sensors, with a mixture of static and mobile sensors, instead of using only mobile sensors as the previous study did.

• We propose a 3-approximation algorithm to solve this NP hard problem for any positively weighted graph.

54P. S.Mandal, IITG

Conclusion

• We have generalized the above problem with different sweep periods and introducing different processing times for the vertices of the graph.

• Our proposed algorithm for this generalized problem achieves approximation factor O(log Ɗ), problem achieves approximation factor O(log Ɗ),

where Ɗ = tmax /tmin , tmin and tmax are the minimum and maximum sweep periods among the vertices.

• If velocities of the mobile sensors are different, we have proved that it is impossible to design any constant factor approximation algorithm to solve the sweep coverage problem unless P=NP.


THANK YOU

56P. S.Mandal, IITG

coverage in wsns (sweep coverage on graph)...sweep coverage sweep coverage let u = {u 1,u 2,…,u n}...

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